score:3
Neither Matthew nor Luke give the time, but He was probably born at night.
The Jews measured a day from sundown to sundown, so "today" is the period from the last sundown to the next sundown.
The announcement is given to the shepherds at night:
8 And there were in the same country shepherds abiding in the field, keeping watch over their flock by night. (Luke 2:8)
The angel says He will be born "this day" (i.e. between last sundown and next sundown):
For unto you is born this day in the city of David a Saviour, which is Christ the Lord. (Luke 2:11)
They went quickly:
And it came to pass, as the angels were gone away from them into heaven, the shepherds said one to another, Let us now go even unto Bethlehem, and see this thing which is come to pass, which the Lord hath made known unto us.
And they came with haste, and found Mary, and Joseph, and the babe lying in a manger. (Luke 2:15-16)
The hills outside of Bethlehem are only a few miles from the (likely) site of Jesus' birth. The shepherds are told during the night that the Savior is/was born "this day", they decide to set out right away, and when the shepherds arrive a short time later, Jesus has already been born. And since He's lying in a manger, not in His mother's arms, the shepherds didn't arrive moments after the birth, but more likely several hours later.
Unless we really stretch the meaning of "haste", the shepherds are there before dawn or very shortly thereafter, and Jesus is already a few hours old.